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Theorem fin23lem30 10382

Description: Lemma for fin23 10429. The residual is disjoint from the common set. (Contributed by Stefan O'Rear, 2-Nov-2014.)

**Hypotheses**
Ref	Expression
fin23lem.a	⊢ 𝑈 = seq_ω((𝑖 ∈ ω, 𝑢 ∈ V ↦ if(((𝑡‘𝑖) ∩ 𝑢) = ∅, 𝑢, ((𝑡‘𝑖) ∩ 𝑢))), ∪ ran 𝑡)
fin23lem17.f	⊢ 𝐹 = {𝑔 ∣ ∀𝑎 ∈ (𝒫 𝑔 ↑_m ω)(∀𝑥 ∈ ω (𝑎‘suc 𝑥) ⊆ (𝑎‘𝑥) → ∩ ran 𝑎 ∈ ran 𝑎)}
fin23lem.b	⊢ 𝑃 = {𝑣 ∈ ω ∣ ∩ ran 𝑈 ⊆ (𝑡‘𝑣)}
fin23lem.c	⊢ 𝑄 = (𝑤 ∈ ω ↦ (℩𝑥 ∈ 𝑃 (𝑥 ∩ 𝑃) ≈ 𝑤))
fin23lem.d	⊢ 𝑅 = (𝑤 ∈ ω ↦ (℩𝑥 ∈ (ω ∖ 𝑃)(𝑥 ∩ (ω ∖ 𝑃)) ≈ 𝑤))
fin23lem.e	⊢ 𝑍 = if(𝑃 ∈ Fin, (𝑡 ∘ 𝑅), ((𝑧 ∈ 𝑃 ↦ ((𝑡‘𝑧) ∖ ∩ ran 𝑈)) ∘ 𝑄))

**Assertion**
Ref	Expression
fin23lem30	⊢ (Fun 𝑡 → (∪ ran 𝑍 ∩ ∩ ran 𝑈) = ∅)

Distinct variable groups: 𝑔,𝑖,𝑡,𝑢,𝑣,𝑥,𝑧,𝑎 𝐹,𝑎,𝑡 𝑤,𝑎,𝑥,𝑧,𝑃 𝑣,𝑎,𝑅,𝑖,𝑢 𝑈,𝑎,𝑖,𝑢,𝑣,𝑧 𝑍,𝑎 𝑔,𝑎 Allowed substitution hints: 𝑃(𝑣,𝑢,𝑡,𝑔,𝑖) 𝑄(𝑥,𝑧,𝑤,𝑣,𝑢,𝑡,𝑔,𝑖,𝑎) 𝑅(𝑥,𝑧,𝑤,𝑡,𝑔) 𝑈(𝑥,𝑤,𝑡,𝑔) 𝐹(𝑥,𝑧,𝑤,𝑣,𝑢,𝑔,𝑖) 𝑍(𝑥,𝑧,𝑤,𝑣,𝑢,𝑡,𝑔,𝑖)

Proof of Theorem fin23lem30 Dummy variable 𝑏 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		fin23lem.e	. 2 ⊢ 𝑍 = if(𝑃 ∈ Fin, (𝑡 ∘ 𝑅), ((𝑧 ∈ 𝑃 ↦ ((𝑡‘𝑧) ∖ ∩ ran 𝑈)) ∘ 𝑄))
2		eqif 4567	. . 3 ⊢ (𝑍 = if(𝑃 ∈ Fin, (𝑡 ∘ 𝑅), ((𝑧 ∈ 𝑃 ↦ ((𝑡‘𝑧) ∖ ∩ ran 𝑈)) ∘ 𝑄)) ↔ ((𝑃 ∈ Fin ∧ 𝑍 = (𝑡 ∘ 𝑅)) ∨ (¬ 𝑃 ∈ Fin ∧ 𝑍 = ((𝑧 ∈ 𝑃 ↦ ((𝑡‘𝑧) ∖ ∩ ran 𝑈)) ∘ 𝑄))))
3	2	biimpi 216	. 2 ⊢ (𝑍 = if(𝑃 ∈ Fin, (𝑡 ∘ 𝑅), ((𝑧 ∈ 𝑃 ↦ ((𝑡‘𝑧) ∖ ∩ ran 𝑈)) ∘ 𝑄)) → ((𝑃 ∈ Fin ∧ 𝑍 = (𝑡 ∘ 𝑅)) ∨ (¬ 𝑃 ∈ Fin ∧ 𝑍 = ((𝑧 ∈ 𝑃 ↦ ((𝑡‘𝑧) ∖ ∩ ran 𝑈)) ∘ 𝑄))))
4		simpr 484	. . . . . . . . . . 11 ⊢ ((𝑃 ∈ Fin ∧ Fun 𝑡) → Fun 𝑡)
5		fin23lem.d	. . . . . . . . . . . 12 ⊢ 𝑅 = (𝑤 ∈ ω ↦ (℩𝑥 ∈ (ω ∖ 𝑃)(𝑥 ∩ (ω ∖ 𝑃)) ≈ 𝑤))
6	5	funmpt2 6605	. . . . . . . . . . 11 ⊢ Fun 𝑅
7		funco 6606	. . . . . . . . . . 11 ⊢ ((Fun 𝑡 ∧ Fun 𝑅) → Fun (𝑡 ∘ 𝑅))
8	4, 6, 7	sylancl 586	. . . . . . . . . 10 ⊢ ((𝑃 ∈ Fin ∧ Fun 𝑡) → Fun (𝑡 ∘ 𝑅))
9		elunirn 7271	. . . . . . . . . 10 ⊢ (Fun (𝑡 ∘ 𝑅) → (𝑎 ∈ ∪ ran (𝑡 ∘ 𝑅) ↔ ∃𝑏 ∈ dom (𝑡 ∘ 𝑅)𝑎 ∈ ((𝑡 ∘ 𝑅)‘𝑏)))
10	8, 9	syl 17	. . . . . . . . 9 ⊢ ((𝑃 ∈ Fin ∧ Fun 𝑡) → (𝑎 ∈ ∪ ran (𝑡 ∘ 𝑅) ↔ ∃𝑏 ∈ dom (𝑡 ∘ 𝑅)𝑎 ∈ ((𝑡 ∘ 𝑅)‘𝑏)))
11		dmcoss 5985	. . . . . . . . . . . 12 ⊢ dom (𝑡 ∘ 𝑅) ⊆ dom 𝑅
12	11	sseli 3979	. . . . . . . . . . 11 ⊢ (𝑏 ∈ dom (𝑡 ∘ 𝑅) → 𝑏 ∈ dom 𝑅)
13		fvco 7007	. . . . . . . . . . . . . . . 16 ⊢ ((Fun 𝑅 ∧ 𝑏 ∈ dom 𝑅) → ((𝑡 ∘ 𝑅)‘𝑏) = (𝑡‘(𝑅‘𝑏)))
14	6, 13	mpan 690	. . . . . . . . . . . . . . 15 ⊢ (𝑏 ∈ dom 𝑅 → ((𝑡 ∘ 𝑅)‘𝑏) = (𝑡‘(𝑅‘𝑏)))
15	14	adantl 481	. . . . . . . . . . . . . 14 ⊢ (((𝑃 ∈ Fin ∧ Fun 𝑡) ∧ 𝑏 ∈ dom 𝑅) → ((𝑡 ∘ 𝑅)‘𝑏) = (𝑡‘(𝑅‘𝑏)))
16	15	eleq2d 2827	. . . . . . . . . . . . 13 ⊢ (((𝑃 ∈ Fin ∧ Fun 𝑡) ∧ 𝑏 ∈ dom 𝑅) → (𝑎 ∈ ((𝑡 ∘ 𝑅)‘𝑏) ↔ 𝑎 ∈ (𝑡‘(𝑅‘𝑏))))
17		incom 4209	. . . . . . . . . . . . . . . 16 ⊢ ((𝑡‘(𝑅‘𝑏)) ∩ ∩ ran 𝑈) = (∩ ran 𝑈 ∩ (𝑡‘(𝑅‘𝑏)))
18		difss 4136	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (ω ∖ 𝑃) ⊆ ω
19		ominf 9294	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ¬ ω ∈ Fin
20		fin23lem.b	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ 𝑃 = {𝑣 ∈ ω ∣ ∩ ran 𝑈 ⊆ (𝑡‘𝑣)}
21	20	ssrab3 4082	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ 𝑃 ⊆ ω
22		undif 4482	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑃 ⊆ ω ↔ (𝑃 ∪ (ω ∖ 𝑃)) = ω)
23	21, 22	mpbi 230	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑃 ∪ (ω ∖ 𝑃)) = ω
24		unfi 9211	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝑃 ∈ Fin ∧ (ω ∖ 𝑃) ∈ Fin) → (𝑃 ∪ (ω ∖ 𝑃)) ∈ Fin)
25	23, 24	eqeltrrid 2846	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝑃 ∈ Fin ∧ (ω ∖ 𝑃) ∈ Fin) → ω ∈ Fin)
26	25	ex 412	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑃 ∈ Fin → ((ω ∖ 𝑃) ∈ Fin → ω ∈ Fin))
27	19, 26	mtoi 199	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑃 ∈ Fin → ¬ (ω ∖ 𝑃) ∈ Fin)
28	27	ad2antrr 726	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝑃 ∈ Fin ∧ Fun 𝑡) ∧ 𝑏 ∈ dom 𝑅) → ¬ (ω ∖ 𝑃) ∈ Fin)
29	5	fin23lem22 10367	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((ω ∖ 𝑃) ⊆ ω ∧ ¬ (ω ∖ 𝑃) ∈ Fin) → 𝑅:ω–1-1-onto→(ω ∖ 𝑃))
30	18, 28, 29	sylancr 587	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝑃 ∈ Fin ∧ Fun 𝑡) ∧ 𝑏 ∈ dom 𝑅) → 𝑅:ω–1-1-onto→(ω ∖ 𝑃))
31		f1of 6848	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑅:ω–1-1-onto→(ω ∖ 𝑃) → 𝑅:ω⟶(ω ∖ 𝑃))
32	30, 31	syl 17	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝑃 ∈ Fin ∧ Fun 𝑡) ∧ 𝑏 ∈ dom 𝑅) → 𝑅:ω⟶(ω ∖ 𝑃))
33		simpr 484	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝑃 ∈ Fin ∧ Fun 𝑡) ∧ 𝑏 ∈ dom 𝑅) → 𝑏 ∈ dom 𝑅)
34	32	fdmd 6746	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝑃 ∈ Fin ∧ Fun 𝑡) ∧ 𝑏 ∈ dom 𝑅) → dom 𝑅 = ω)
35	33, 34	eleqtrd 2843	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝑃 ∈ Fin ∧ Fun 𝑡) ∧ 𝑏 ∈ dom 𝑅) → 𝑏 ∈ ω)
36	32, 35	ffvelcdmd 7105	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑃 ∈ Fin ∧ Fun 𝑡) ∧ 𝑏 ∈ dom 𝑅) → (𝑅‘𝑏) ∈ (ω ∖ 𝑃))
37	36	eldifbd 3964	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑃 ∈ Fin ∧ Fun 𝑡) ∧ 𝑏 ∈ dom 𝑅) → ¬ (𝑅‘𝑏) ∈ 𝑃)
38	20	eleq2i 2833	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑅‘𝑏) ∈ 𝑃 ↔ (𝑅‘𝑏) ∈ {𝑣 ∈ ω ∣ ∩ ran 𝑈 ⊆ (𝑡‘𝑣)})
39	37, 38	sylnib 328	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑃 ∈ Fin ∧ Fun 𝑡) ∧ 𝑏 ∈ dom 𝑅) → ¬ (𝑅‘𝑏) ∈ {𝑣 ∈ ω ∣ ∩ ran 𝑈 ⊆ (𝑡‘𝑣)})
40	36	eldifad 3963	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑃 ∈ Fin ∧ Fun 𝑡) ∧ 𝑏 ∈ dom 𝑅) → (𝑅‘𝑏) ∈ ω)
41		fveq2 6906	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑣 = (𝑅‘𝑏) → (𝑡‘𝑣) = (𝑡‘(𝑅‘𝑏)))
42	41	sseq2d 4016	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑣 = (𝑅‘𝑏) → (∩ ran 𝑈 ⊆ (𝑡‘𝑣) ↔ ∩ ran 𝑈 ⊆ (𝑡‘(𝑅‘𝑏))))
43	42	elrab3 3693	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑅‘𝑏) ∈ ω → ((𝑅‘𝑏) ∈ {𝑣 ∈ ω ∣ ∩ ran 𝑈 ⊆ (𝑡‘𝑣)} ↔ ∩ ran 𝑈 ⊆ (𝑡‘(𝑅‘𝑏))))
44	40, 43	syl 17	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑃 ∈ Fin ∧ Fun 𝑡) ∧ 𝑏 ∈ dom 𝑅) → ((𝑅‘𝑏) ∈ {𝑣 ∈ ω ∣ ∩ ran 𝑈 ⊆ (𝑡‘𝑣)} ↔ ∩ ran 𝑈 ⊆ (𝑡‘(𝑅‘𝑏))))
45	39, 44	mtbid 324	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑃 ∈ Fin ∧ Fun 𝑡) ∧ 𝑏 ∈ dom 𝑅) → ¬ ∩ ran 𝑈 ⊆ (𝑡‘(𝑅‘𝑏)))
46		fin23lem.a	. . . . . . . . . . . . . . . . . . 19 ⊢ 𝑈 = seq_ω((𝑖 ∈ ω, 𝑢 ∈ V ↦ if(((𝑡‘𝑖) ∩ 𝑢) = ∅, 𝑢, ((𝑡‘𝑖) ∩ 𝑢))), ∪ ran 𝑡)
47	46	fin23lem20 10377	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑅‘𝑏) ∈ ω → (∩ ran 𝑈 ⊆ (𝑡‘(𝑅‘𝑏)) ∨ (∩ ran 𝑈 ∩ (𝑡‘(𝑅‘𝑏))) = ∅))
48	40, 47	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑃 ∈ Fin ∧ Fun 𝑡) ∧ 𝑏 ∈ dom 𝑅) → (∩ ran 𝑈 ⊆ (𝑡‘(𝑅‘𝑏)) ∨ (∩ ran 𝑈 ∩ (𝑡‘(𝑅‘𝑏))) = ∅))
49		orel1 889	. . . . . . . . . . . . . . . . 17 ⊢ (¬ ∩ ran 𝑈 ⊆ (𝑡‘(𝑅‘𝑏)) → ((∩ ran 𝑈 ⊆ (𝑡‘(𝑅‘𝑏)) ∨ (∩ ran 𝑈 ∩ (𝑡‘(𝑅‘𝑏))) = ∅) → (∩ ran 𝑈 ∩ (𝑡‘(𝑅‘𝑏))) = ∅))
50	45, 48, 49	sylc 65	. . . . . . . . . . . . . . . 16 ⊢ (((𝑃 ∈ Fin ∧ Fun 𝑡) ∧ 𝑏 ∈ dom 𝑅) → (∩ ran 𝑈 ∩ (𝑡‘(𝑅‘𝑏))) = ∅)
51	17, 50	eqtrid 2789	. . . . . . . . . . . . . . 15 ⊢ (((𝑃 ∈ Fin ∧ Fun 𝑡) ∧ 𝑏 ∈ dom 𝑅) → ((𝑡‘(𝑅‘𝑏)) ∩ ∩ ran 𝑈) = ∅)
52		disj 4450	. . . . . . . . . . . . . . 15 ⊢ (((𝑡‘(𝑅‘𝑏)) ∩ ∩ ran 𝑈) = ∅ ↔ ∀𝑎 ∈ (𝑡‘(𝑅‘𝑏)) ¬ 𝑎 ∈ ∩ ran 𝑈)
53	51, 52	sylib 218	. . . . . . . . . . . . . 14 ⊢ (((𝑃 ∈ Fin ∧ Fun 𝑡) ∧ 𝑏 ∈ dom 𝑅) → ∀𝑎 ∈ (𝑡‘(𝑅‘𝑏)) ¬ 𝑎 ∈ ∩ ran 𝑈)
54		rsp 3247	. . . . . . . . . . . . . 14 ⊢ (∀𝑎 ∈ (𝑡‘(𝑅‘𝑏)) ¬ 𝑎 ∈ ∩ ran 𝑈 → (𝑎 ∈ (𝑡‘(𝑅‘𝑏)) → ¬ 𝑎 ∈ ∩ ran 𝑈))
55	53, 54	syl 17	. . . . . . . . . . . . 13 ⊢ (((𝑃 ∈ Fin ∧ Fun 𝑡) ∧ 𝑏 ∈ dom 𝑅) → (𝑎 ∈ (𝑡‘(𝑅‘𝑏)) → ¬ 𝑎 ∈ ∩ ran 𝑈))
56	16, 55	sylbid 240	. . . . . . . . . . . 12 ⊢ (((𝑃 ∈ Fin ∧ Fun 𝑡) ∧ 𝑏 ∈ dom 𝑅) → (𝑎 ∈ ((𝑡 ∘ 𝑅)‘𝑏) → ¬ 𝑎 ∈ ∩ ran 𝑈))
57	56	ex 412	. . . . . . . . . . 11 ⊢ ((𝑃 ∈ Fin ∧ Fun 𝑡) → (𝑏 ∈ dom 𝑅 → (𝑎 ∈ ((𝑡 ∘ 𝑅)‘𝑏) → ¬ 𝑎 ∈ ∩ ran 𝑈)))
58	12, 57	syl5 34	. . . . . . . . . 10 ⊢ ((𝑃 ∈ Fin ∧ Fun 𝑡) → (𝑏 ∈ dom (𝑡 ∘ 𝑅) → (𝑎 ∈ ((𝑡 ∘ 𝑅)‘𝑏) → ¬ 𝑎 ∈ ∩ ran 𝑈)))
59	58	rexlimdv 3153	. . . . . . . . 9 ⊢ ((𝑃 ∈ Fin ∧ Fun 𝑡) → (∃𝑏 ∈ dom (𝑡 ∘ 𝑅)𝑎 ∈ ((𝑡 ∘ 𝑅)‘𝑏) → ¬ 𝑎 ∈ ∩ ran 𝑈))
60	10, 59	sylbid 240	. . . . . . . 8 ⊢ ((𝑃 ∈ Fin ∧ Fun 𝑡) → (𝑎 ∈ ∪ ran (𝑡 ∘ 𝑅) → ¬ 𝑎 ∈ ∩ ran 𝑈))
61	60	ralrimiv 3145	. . . . . . 7 ⊢ ((𝑃 ∈ Fin ∧ Fun 𝑡) → ∀𝑎 ∈ ∪ ran (𝑡 ∘ 𝑅) ¬ 𝑎 ∈ ∩ ran 𝑈)
62		disj 4450	. . . . . . 7 ⊢ ((∪ ran (𝑡 ∘ 𝑅) ∩ ∩ ran 𝑈) = ∅ ↔ ∀𝑎 ∈ ∪ ran (𝑡 ∘ 𝑅) ¬ 𝑎 ∈ ∩ ran 𝑈)
63	61, 62	sylibr 234	. . . . . 6 ⊢ ((𝑃 ∈ Fin ∧ Fun 𝑡) → (∪ ran (𝑡 ∘ 𝑅) ∩ ∩ ran 𝑈) = ∅)
64		rneq 5947	. . . . . . . . 9 ⊢ (𝑍 = (𝑡 ∘ 𝑅) → ran 𝑍 = ran (𝑡 ∘ 𝑅))
65	64	unieqd 4920	. . . . . . . 8 ⊢ (𝑍 = (𝑡 ∘ 𝑅) → ∪ ran 𝑍 = ∪ ran (𝑡 ∘ 𝑅))
66	65	ineq1d 4219	. . . . . . 7 ⊢ (𝑍 = (𝑡 ∘ 𝑅) → (∪ ran 𝑍 ∩ ∩ ran 𝑈) = (∪ ran (𝑡 ∘ 𝑅) ∩ ∩ ran 𝑈))
67	66	eqeq1d 2739	. . . . . 6 ⊢ (𝑍 = (𝑡 ∘ 𝑅) → ((∪ ran 𝑍 ∩ ∩ ran 𝑈) = ∅ ↔ (∪ ran (𝑡 ∘ 𝑅) ∩ ∩ ran 𝑈) = ∅))
68	63, 67	imbitrrid 246	. . . . 5 ⊢ (𝑍 = (𝑡 ∘ 𝑅) → ((𝑃 ∈ Fin ∧ Fun 𝑡) → (∪ ran 𝑍 ∩ ∩ ran 𝑈) = ∅))
69	68	expd 415	. . . 4 ⊢ (𝑍 = (𝑡 ∘ 𝑅) → (𝑃 ∈ Fin → (Fun 𝑡 → (∪ ran 𝑍 ∩ ∩ ran 𝑈) = ∅)))
70	69	impcom 407	. . 3 ⊢ ((𝑃 ∈ Fin ∧ 𝑍 = (𝑡 ∘ 𝑅)) → (Fun 𝑡 → (∪ ran 𝑍 ∩ ∩ ran 𝑈) = ∅))
71		rneq 5947	. . . . . . . 8 ⊢ (𝑍 = ((𝑧 ∈ 𝑃 ↦ ((𝑡‘𝑧) ∖ ∩ ran 𝑈)) ∘ 𝑄) → ran 𝑍 = ran ((𝑧 ∈ 𝑃 ↦ ((𝑡‘𝑧) ∖ ∩ ran 𝑈)) ∘ 𝑄))
72	71	unieqd 4920	. . . . . . 7 ⊢ (𝑍 = ((𝑧 ∈ 𝑃 ↦ ((𝑡‘𝑧) ∖ ∩ ran 𝑈)) ∘ 𝑄) → ∪ ran 𝑍 = ∪ ran ((𝑧 ∈ 𝑃 ↦ ((𝑡‘𝑧) ∖ ∩ ran 𝑈)) ∘ 𝑄))
73	72	ineq1d 4219	. . . . . 6 ⊢ (𝑍 = ((𝑧 ∈ 𝑃 ↦ ((𝑡‘𝑧) ∖ ∩ ran 𝑈)) ∘ 𝑄) → (∪ ran 𝑍 ∩ ∩ ran 𝑈) = (∪ ran ((𝑧 ∈ 𝑃 ↦ ((𝑡‘𝑧) ∖ ∩ ran 𝑈)) ∘ 𝑄) ∩ ∩ ran 𝑈))
74		rncoss 5986	. . . . . . . 8 ⊢ ran ((𝑧 ∈ 𝑃 ↦ ((𝑡‘𝑧) ∖ ∩ ran 𝑈)) ∘ 𝑄) ⊆ ran (𝑧 ∈ 𝑃 ↦ ((𝑡‘𝑧) ∖ ∩ ran 𝑈))
75	74	unissi 4916	. . . . . . 7 ⊢ ∪ ran ((𝑧 ∈ 𝑃 ↦ ((𝑡‘𝑧) ∖ ∩ ran 𝑈)) ∘ 𝑄) ⊆ ∪ ran (𝑧 ∈ 𝑃 ↦ ((𝑡‘𝑧) ∖ ∩ ran 𝑈))
76		disj 4450	. . . . . . . 8 ⊢ ((∪ ran (𝑧 ∈ 𝑃 ↦ ((𝑡‘𝑧) ∖ ∩ ran 𝑈)) ∩ ∩ ran 𝑈) = ∅ ↔ ∀𝑎 ∈ ∪ ran (𝑧 ∈ 𝑃 ↦ ((𝑡‘𝑧) ∖ ∩ ran 𝑈)) ¬ 𝑎 ∈ ∩ ran 𝑈)
77		eluniab 4921	. . . . . . . . . 10 ⊢ (𝑎 ∈ ∪ {𝑏 ∣ ∃𝑧 ∈ 𝑃 𝑏 = ((𝑡‘𝑧) ∖ ∩ ran 𝑈)} ↔ ∃𝑏(𝑎 ∈ 𝑏 ∧ ∃𝑧 ∈ 𝑃 𝑏 = ((𝑡‘𝑧) ∖ ∩ ran 𝑈)))
78		eleq2 2830	. . . . . . . . . . . . . 14 ⊢ (𝑏 = ((𝑡‘𝑧) ∖ ∩ ran 𝑈) → (𝑎 ∈ 𝑏 ↔ 𝑎 ∈ ((𝑡‘𝑧) ∖ ∩ ran 𝑈)))
79		eldifn 4132	. . . . . . . . . . . . . 14 ⊢ (𝑎 ∈ ((𝑡‘𝑧) ∖ ∩ ran 𝑈) → ¬ 𝑎 ∈ ∩ ran 𝑈)
80	78, 79	biimtrdi 253	. . . . . . . . . . . . 13 ⊢ (𝑏 = ((𝑡‘𝑧) ∖ ∩ ran 𝑈) → (𝑎 ∈ 𝑏 → ¬ 𝑎 ∈ ∩ ran 𝑈))
81	80	rexlimivw 3151	. . . . . . . . . . . 12 ⊢ (∃𝑧 ∈ 𝑃 𝑏 = ((𝑡‘𝑧) ∖ ∩ ran 𝑈) → (𝑎 ∈ 𝑏 → ¬ 𝑎 ∈ ∩ ran 𝑈))
82	81	impcom 407	. . . . . . . . . . 11 ⊢ ((𝑎 ∈ 𝑏 ∧ ∃𝑧 ∈ 𝑃 𝑏 = ((𝑡‘𝑧) ∖ ∩ ran 𝑈)) → ¬ 𝑎 ∈ ∩ ran 𝑈)
83	82	exlimiv 1930	. . . . . . . . . 10 ⊢ (∃𝑏(𝑎 ∈ 𝑏 ∧ ∃𝑧 ∈ 𝑃 𝑏 = ((𝑡‘𝑧) ∖ ∩ ran 𝑈)) → ¬ 𝑎 ∈ ∩ ran 𝑈)
84	77, 83	sylbi 217	. . . . . . . . 9 ⊢ (𝑎 ∈ ∪ {𝑏 ∣ ∃𝑧 ∈ 𝑃 𝑏 = ((𝑡‘𝑧) ∖ ∩ ran 𝑈)} → ¬ 𝑎 ∈ ∩ ran 𝑈)
85		eqid 2737	. . . . . . . . . . 11 ⊢ (𝑧 ∈ 𝑃 ↦ ((𝑡‘𝑧) ∖ ∩ ran 𝑈)) = (𝑧 ∈ 𝑃 ↦ ((𝑡‘𝑧) ∖ ∩ ran 𝑈))
86	85	rnmpt 5968	. . . . . . . . . 10 ⊢ ran (𝑧 ∈ 𝑃 ↦ ((𝑡‘𝑧) ∖ ∩ ran 𝑈)) = {𝑏 ∣ ∃𝑧 ∈ 𝑃 𝑏 = ((𝑡‘𝑧) ∖ ∩ ran 𝑈)}
87	86	unieqi 4919	. . . . . . . . 9 ⊢ ∪ ran (𝑧 ∈ 𝑃 ↦ ((𝑡‘𝑧) ∖ ∩ ran 𝑈)) = ∪ {𝑏 ∣ ∃𝑧 ∈ 𝑃 𝑏 = ((𝑡‘𝑧) ∖ ∩ ran 𝑈)}
88	84, 87	eleq2s 2859	. . . . . . . 8 ⊢ (𝑎 ∈ ∪ ran (𝑧 ∈ 𝑃 ↦ ((𝑡‘𝑧) ∖ ∩ ran 𝑈)) → ¬ 𝑎 ∈ ∩ ran 𝑈)
89	76, 88	mprgbir 3068	. . . . . . 7 ⊢ (∪ ran (𝑧 ∈ 𝑃 ↦ ((𝑡‘𝑧) ∖ ∩ ran 𝑈)) ∩ ∩ ran 𝑈) = ∅
90		ssdisj 4460	. . . . . . 7 ⊢ ((∪ ran ((𝑧 ∈ 𝑃 ↦ ((𝑡‘𝑧) ∖ ∩ ran 𝑈)) ∘ 𝑄) ⊆ ∪ ran (𝑧 ∈ 𝑃 ↦ ((𝑡‘𝑧) ∖ ∩ ran 𝑈)) ∧ (∪ ran (𝑧 ∈ 𝑃 ↦ ((𝑡‘𝑧) ∖ ∩ ran 𝑈)) ∩ ∩ ran 𝑈) = ∅) → (∪ ran ((𝑧 ∈ 𝑃 ↦ ((𝑡‘𝑧) ∖ ∩ ran 𝑈)) ∘ 𝑄) ∩ ∩ ran 𝑈) = ∅)
91	75, 89, 90	mp2an 692	. . . . . 6 ⊢ (∪ ran ((𝑧 ∈ 𝑃 ↦ ((𝑡‘𝑧) ∖ ∩ ran 𝑈)) ∘ 𝑄) ∩ ∩ ran 𝑈) = ∅
92	73, 91	eqtrdi 2793	. . . . 5 ⊢ (𝑍 = ((𝑧 ∈ 𝑃 ↦ ((𝑡‘𝑧) ∖ ∩ ran 𝑈)) ∘ 𝑄) → (∪ ran 𝑍 ∩ ∩ ran 𝑈) = ∅)
93	92	a1d 25	. . . 4 ⊢ (𝑍 = ((𝑧 ∈ 𝑃 ↦ ((𝑡‘𝑧) ∖ ∩ ran 𝑈)) ∘ 𝑄) → (Fun 𝑡 → (∪ ran 𝑍 ∩ ∩ ran 𝑈) = ∅))
94	93	adantl 481	. . 3 ⊢ ((¬ 𝑃 ∈ Fin ∧ 𝑍 = ((𝑧 ∈ 𝑃 ↦ ((𝑡‘𝑧) ∖ ∩ ran 𝑈)) ∘ 𝑄)) → (Fun 𝑡 → (∪ ran 𝑍 ∩ ∩ ran 𝑈) = ∅))
95	70, 94	jaoi 858	. 2 ⊢ (((𝑃 ∈ Fin ∧ 𝑍 = (𝑡 ∘ 𝑅)) ∨ (¬ 𝑃 ∈ Fin ∧ 𝑍 = ((𝑧 ∈ 𝑃 ↦ ((𝑡‘𝑧) ∖ ∩ ran 𝑈)) ∘ 𝑄))) → (Fun 𝑡 → (∪ ran 𝑍 ∩ ∩ ran 𝑈) = ∅))
96	1, 3, 95	mp2b 10	1 ⊢ (Fun 𝑡 → (∪ ran 𝑍 ∩ ∩ ran 𝑈) = ∅)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 206 ∧ wa 395 ∨ wo 848 = wceq 1540 ∃wex 1779 ∈ wcel 2108 {cab 2714 ∀wral 3061 ∃wrex 3070 {crab 3436 Vcvv 3480 ∖ cdif 3948 ∪ cun 3949 ∩ cin 3950 ⊆ wss 3951 ∅c0 4333 ifcif 4525 𝒫 cpw 4600 ∪ cuni 4907 ∩ cint 4946 class class class wbr 5143 ↦ cmpt 5225 dom cdm 5685 ran crn 5686 ∘ ccom 5689 suc csuc 6386 Fun wfun 6555 ⟶wf 6557 –1-1-onto→wf1o 6560 ‘cfv 6561 ℩crio 7387 (class class class)co 7431 ∈ cmpo 7433 ωcom 7887 seq_ωcseqom 8487 ↑_m cmap 8866 ≈ cen 8982 Fincfn 8985

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-int 4947 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-se 5638 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-isom 6570 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-2nd 8015 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-seqom 8488 df-1o 8506 df-er 8745 df-en 8986 df-dom 8987 df-sdom 8988 df-fin 8989 df-card 9979

This theorem is referenced by: fin23lem31 10383

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